Calculating Energy of Sequence x(n) with u(n-1)

[caramello]

Hi, I'm sorry if I post this question at the wrong section. I have a question regarding the energy sequence.

Qn: find the energy of the sequence x(n) = (1/3)^n u(n-1) + (1/2)^(n-1) u(n-2)

My Approach:
1). Energy = (sum from negative infinity to infinity) of |x(n)|^2
2). I know that if a sequence is in the form of x(n) = a r^n u(n), for example if x(n) = (1/3)^n then we can compute the energy simply by E = (sum from 0 to infinity) of (1/3)^n = 1/(1-1/3) = 3/2.

My question is:
1). What if it is x(n) = a r^(n-1) instead of only x(n) = a r^n
2). And when u(n-1) instead of u(n)
*what I knew for u(n-1) is that we need to consider the sum from 1 to infinity instead of from 0 to infinity(like when it's only u(n)), but then after that I'm not sure on how to start the calculation at all. I'm so confused about this.

Does anyone can help me with this?

Thank you!

 

[KataKoniK]

Hello,

Thank you for your question. I am happy to assist you with finding the energy of the given sequence.

Firstly, let's define the sequence x(n) as:

x(n) = (1/3)^n u(n-1) + (1/2)^(n-1) u(n-2)

To find the energy of this sequence, we can use the formula E = (sum from negative infinity to infinity) of |x(n)|^2.

Now, let's break down the sequence into two parts:

x(n) = (1/3)^n u(n-1) + (1/2)^(n-1) u(n-2)

= (1/3)^n * u(n-1) + (1/2)^n * (1/2)^(-1) * u(n-1)

= (1/3)^n * u(n-1) + (1/2)^n * (1/2)^n * u(n-1)

= (1/3)^n * u(n-1) + (1/2)^n * (1/2)^n * u(n-1)

= (1/3)^n * u(n-1) + (1/2)^n * u(n-1)

= (1/3)^n * u(n-1) + (1/2)^n * u(n-1)

Now, we can use the formula E = (sum from negative infinity to infinity) of |x(n)|^2 to calculate the energy of the sequence.

E = (sum from negative infinity to infinity) of |(1/3)^n * u(n-1) + (1/2)^n * u(n-1)|^2

= (sum from negative infinity to infinity) of |(1/3)^n|^2 * |u(n-1)|^2 + |(1/2)^n|^2 * |u(n-1)|^2

= (sum from negative infinity to infinity) of (1/9)^n * |u(n-1)|^2 + (1/4)^n * |u(n-1)|^2

= (sum from negative infinity to infinity) of [(1/9)^n + (1/4)^n] * |u(n-1)|^2

= (sum from negative

 

[Mene]

Hello there,

Calculating the energy of a sequence can be a bit tricky, but fortunately there are some clear steps you can follow to make it easier.

First, let's define the sequence x(n) = (1/3)^n u(n-1) + (1/2)^(n-1) u(n-2). In this case, we have two terms in the sequence, one with a coefficient of (1/3)^n and the other with a coefficient of (1/2)^(n-1).

To calculate the energy of this sequence, we can use your approach of taking the sum from negative infinity to infinity of |x(n)|^2. However, we need to account for both terms in the sequence separately.

For the first term, (1/3)^n u(n-1), we can use the formula you mentioned for a sequence in the form of x(n) = a r^n u(n). In this case, a = 1/3 and r = 1/3. So the energy for this term would be E1 = (sum from 0 to infinity) of |(1/3)^n|^2 = (sum from 0 to infinity) of (1/3)^(2n) = (1/(1-1/9)) = 9/8.

For the second term, (1/2)^(n-1) u(n-2), we can use the same formula but with a = 1/2 and r = 1/2. So the energy for this term would be E2 = (sum from 0 to infinity) of |(1/2)^(n-1)|^2 = (sum from 0 to infinity) of (1/2)^(2n-2) = (1/(1-1/4)) = 4/3.

Now, to get the total energy of the sequence, we simply add these two energies together: E = E1 + E2 = 9/8 + 4/3 = 21/8.

To answer your questions, if the sequence is in the form of x(n) = a r^(n-1), we can still use the same formula for the energy, but we need to adjust the limits of the sum accordingly. For u(n-1) instead of u(n), we need to consider the sum from 1